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In <3811@deimos.cis.ksu.edu> mac@harris.cis.ksu.edu (Myron A. Calhoun) writes:
>I'm under the impression that in any given range which doesn't have
>singularity points, Chebyshev polynomials (of the first kind) can be
>used to "telescope" Taylor series of degree N down to degree N-1
>while retaining the same maximum error (but this error may pop up in
>places where it wasn't present in the original--kinda like squeezing
>a balloon).

Well, it was extreme of me to call Taylor series wimpy, especially since
sin  is  about   the best example   of a  Taylor  --   it's  entire  (no
singularities)  and the   coefficients decrease fast.    Also,  a Taylor
series is often by far the easiest approximation to a function.

However, even with sin, Chebyshev have an advantage of more than 1 term,
and the advantage increases with the degree.   Consider sin(x) from 0 to
pi/2 -- the obvious interval for a  sin implementation.   I expanded the
Taylor series about pi/4, and  did the Chebyshev  over [0,pi/2], both to
20 significant  digits.  These coefs are  the approx  max  error at each
stage since I also renormalized x to be in [-1,1].

   n abs(Taylor coef) abs(Cheb coef (i.e. coef of T(n,x)))
   5 2e-3        7e-6
   10 2e-8	 3e-11
   15 1e-14	 9e-19

The 15th degree Taylor has an accuracy  between  the 12 and  13th degree
Chebyshev -  more  than 1   degree  improvement.  For  other series   or
intervals, it can be more.  Consider sin(x) from -pi  to pi.  I expanded
sin(x)/x  about 0.  Note that  it's an  even function so there's  no 5th
degree.

   n Taylor Chebyshev
   4 3      .2
   10 7e-3 1e-5
   14 2e-5 2e-9

The 14th degree Taylor is between an 8th and 10th degree Chebyshev.

Next,  your restriction to non singular  functions is quite strict since
many smooth, infinitely differentiable, functions, such as 1/(x^2+0.01),
have a radius  of convergence for  the  Taylor  of  0.1  because of  the
complex pole  at 0+.1i.  Also,  an N-th degree  Taylor requires that the
function be differentiable  N   times, whereas  Chebyshev  requires only
once.  

In both  of these cases,  the  Chebyshev series converges slowly, but it
converges over the whole   interval,  whereas  Taylor converges  only in
(-0.1, 0.1) in the first, and not at all in the second.

Actually, if you allow one division in your approximation, you can use a
rational or Pade form of either the Taylor or Chebyshev, which  for less
well behaved functions vastly improves either  approx. They're much more
complex to calculate, but can improve on the Taylor even for exp(x).

Summary: Taylor is certainly ok for sin(x), but Chebyshev is faster even
here, and does a much wider class of functions.  However,  it is messier
to calculate.


-- 
						   Wm. Randolph Franklin
Internet: wrf@ecse.rpi.edu (or @cs.rpi.edu)    Bitnet: Wrfrankl@Rpitsmts
Telephone: (518) 276-6077;  Telex: 6716050 RPI TROU; Fax: (518) 276-6261
Paper: ECSE Dept., 6026 JEC, Rensselaer Polytechnic Inst, Troy NY, 12180